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Title: Quasi-periodic solutions of nonlinear beam equation with prescribed frequencies

Abstract

Consider the one dimensional nonlinear beam equation u{sub tt} + u{sub xxxx} + mu + u{sup 3} = 0 under Dirichlet boundary conditions. We show that for any m > 0 but a set of small Lebesgue measure, the above equation admits a family of small-amplitude quasi-periodic solutions with n-dimensional Diophantine frequencies. These Diophantine frequencies are the small dilation of a prescribed Diophantine vector. The proofs are based on an infinite dimensional Kolmogorov-Arnold-Moser iteration procedure and a partial Birkhoff normal form. .

Authors:
 [1]; ;  [2]
  1. College of Information Technology, Jilin Agricultural University, Changchun 130118 (China)
  2. School of Mathematics and Statistics, and Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun 130024 (China)
Publication Date:
OSTI Identifier:
22403146
Resource Type:
Journal Article
Journal Name:
Journal of Mathematical Physics
Additional Journal Information:
Journal Volume: 56; Journal Issue: 5; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0022-2488
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; BEAMS; DIRICHLET PROBLEM; EQUATIONS; MATHEMATICAL SOLUTIONS; NONLINEAR PROBLEMS; ONE-DIMENSIONAL CALCULATIONS; PERIODICITY

Citation Formats

Chang, Jing, Gao, Yixian, E-mail: gaoyx643@nenu.edu.cn, and Li, Yong. Quasi-periodic solutions of nonlinear beam equation with prescribed frequencies. United States: N. p., 2015. Web. doi:10.1063/1.4919673.
Chang, Jing, Gao, Yixian, E-mail: gaoyx643@nenu.edu.cn, & Li, Yong. Quasi-periodic solutions of nonlinear beam equation with prescribed frequencies. United States. doi:10.1063/1.4919673.
Chang, Jing, Gao, Yixian, E-mail: gaoyx643@nenu.edu.cn, and Li, Yong. Fri . "Quasi-periodic solutions of nonlinear beam equation with prescribed frequencies". United States. doi:10.1063/1.4919673.
@article{osti_22403146,
title = {Quasi-periodic solutions of nonlinear beam equation with prescribed frequencies},
author = {Chang, Jing and Gao, Yixian, E-mail: gaoyx643@nenu.edu.cn and Li, Yong},
abstractNote = {Consider the one dimensional nonlinear beam equation u{sub tt} + u{sub xxxx} + mu + u{sup 3} = 0 under Dirichlet boundary conditions. We show that for any m > 0 but a set of small Lebesgue measure, the above equation admits a family of small-amplitude quasi-periodic solutions with n-dimensional Diophantine frequencies. These Diophantine frequencies are the small dilation of a prescribed Diophantine vector. The proofs are based on an infinite dimensional Kolmogorov-Arnold-Moser iteration procedure and a partial Birkhoff normal form. .},
doi = {10.1063/1.4919673},
journal = {Journal of Mathematical Physics},
issn = {0022-2488},
number = 5,
volume = 56,
place = {United States},
year = {2015},
month = {5}
}